1

Roumen Tsekov

Department of Physical Chemistry, University of Sofia, 1164 Sofia, Bulgaria

Abstract

Due to limitation of the binding energy of a self-gravitating matter, the ra-dius of a body is at least twice larger than the Schwarzschild rara-dius. The total en-ergy is adsorbed at the body surface, giving rise of a size-dependent surface ten-sion. Since the Hawking temperature appears to be the critical one, the black holes possess zero surface tension. Microscopic neutrino stars are also introduced.

Keywords: self-gravitation, black holes, neutrino stars.

A black hole is so condensed that even light cannot escape due to the gravitational attrac-tion. Leaving apart any resistance due to particle repulsion and exclusion the black hole formation seems also energetically impossible. Imagine a self-gravitating star, emitting radiation in all. The kinetic energy of the matter particles is continuously decreasing and the star is shrinking towards a black hole. However, due to the mass-energy equivalence, the star is losing its mass as well, thus reducing the inner strength of the self-gravitational attraction. As the Schwarzschild radius decreases also with decreasing mass, this Zeno effect leads to the conclusion that only an empty black hole could form with zero size, when the mass of the star becomes zero.

To make the picture consistent, let us start from the Newton gravity. If the mass density

is radial-symmetrically distributed, the corresponding gravitational potential obeys the Pois-son equation

2 2

( ) / 4

*r* *r* *r* *r* *G*

2

/
*r* *Gmr* *r*

(1)

The second integral form involves the mass 2 0 4

*r*
*r*

*m*

##

*r dr*, enclosed within a central sphere with radius

*r*. Naturally,

*m*

_{}contains the total mass

*M*0 of the self-gravitating matter. Using Eq. (1), one can calculate the overall potential energy

2 2

0 2 ( / 2) 0 ( *r* / )

*U*

##

*r dr*

*G*

##

*m*

*r dr*(2)

2

Since no other interactions are considered and the excess energy is freely released as radiation
in vacuum, the body contracts permanently due to the gravitational attraction. At the end, a mass
point is formed with *mr* *M*0 everywhere. According to Eq. (1), it corresponds to the classical
Newton potential *GM*0/*r*. As it is well known, U diverges for a mass point, which indicates
a generic problem of the Newton gravity.

According to the Einstein special relativity theory, the total energy at rest 2
0
*E**M c* *U*

is positively defined. The lowest value *E*0 limits the binding energy to the maximal 100 % mass
defect. The integral and differential forms of 2

0 /

*U c* *M*

reads

2

0 0

(*G*/ 2)

##

(*m*/

_{r}*cr dr*)

##

(*) 2 ( / 2)( / )*

_{r}m dr_{r}*rmr*

*G*

*mr*

*cr*

(3)

The integration Eq. (3) leads straightforward to the radial mass distribution

0/ (1 0/ )
*r*

*m* *M* *R* *r* 2

0 0/ 2

*R* *GM* *c* (4)

which is not constant anymore due to the binging energy limitation. Substituting Eq. (4) in Eq. (1) yields after integration the restricted gravitational potential

2

0
2*c* ln(1 *R* / )*r*

(5)

Far from the center it tends to the Newton potential, while near the center the logarithmic po-tential 2

0

2*c* ln( /*r R* ) is much weaker than *GM*0 /*r*. That is why the potential energy (2) is finite.
One can derive the mass density distribution by differentiating directly *mr* from Eq. (4)

2 2

0 0/ 4 ( 0)
*M R* *r r* *R*

(6)

If the mass *M*0 is very small, Eq. (4) approaches a mass point as expected. In the opposite
case of large masses, Eqs. (4) and (6) tends to *mr* *M r R*0( / 0) and

2
0/ 4 0
*M* *R r*

3

density 0 is zero and *mr* *M*0. One can calculate the corresponding pressure *p* via the
hy-drostatic force balance

2 /

*rp* *r* *Gmr* *r*

(7)

It is zero outside the body but remarkably 2

*p* *c* resembles inside an equation of state. As is
seen, the limitation of the binding energy resolves the mass point singularity. However, since *R*0
is one fourth of the Schwarzschild radius, a black hole singularity still holds. It points out that
probably the maximal mass will prevent such a peculiarity as well.

It is interesting what the effect would be of the binding energy limitation on black holes. To answer this question, we are going to repeat the analysis above, using the Einstein general relativity theory. In the frames of the latter, the problem is described by the Tolman-Oppenhei-mer-Volkoff (TOV) equations1

3 2 2 2

( 4 / ) / (1 2 / )
*r* *G mr* *r p c* *r* *Gmr* *c r*

2

( / )

*rp* *p c* *r*

(8)

from which Eqs. (1) and (7) follow, respectively, in the non-relativistic limit. Note that the
relativ-istic mass *M* is smaller than *M*0 and the difference

2

0 /

*M* *M* *U c* is the binding energy mass
defect. For a mass point, Eq. (8) reduces to 2 2

/ (1 2 / )
*r* *GM r* *GM c r*

, since *mr* *M* , 0

and *p*0. Integrating this equation results straightforward in the well-known relativistic

poten-tial, where 2

2 /

*s*

*r* *GM c* is the Schwarzschild radius,2

2

2 / *c* ln(1*r _{s}*/ )

*r*(9)

Far from the center, tends to the Newton potential *GM r*/ . The singularity at *rs* marks the
event horizon,3_{ where the surface of the black hole takes place. }

Let us apply now the energy limitation to the TOV equations (8). In general, the pressure
*p* *X* is proportional to the energy density 2

*c*

and we got *X* 1 in the semi-relativistic
analysis above. Introducing 2

4

2 2

( ) / (1 2 / )

*r* *G mr* *Xr mr* *r* *r* *Gmr* *c r*

2

(1 )

*r* *r*

*Xc* *X* (10)

Searching for a black hole, we are looking for a compact body of self-gravitating matter. For radii
larger than the body radius *R* the standard expressions *mr* *M* and 0 hold. It follows
im-mediately from Eq. (10) that Eq. (9) is the potential outside the body. It is well known that inside
the body *mr* *M r R*( / ) and

2

/ 4

*M* *Rr*

are solutions of Eq. (10),4_{ which is also supported by }

our semi-relativistic analysis. Introducing them in Eq. (10) leads to

2
/ * _{s}* 1 (1 ) / 4

*R r* *X* *X* * _{R}* 2

*c*2ln( / )

*r R X*/ (1

*X*) (11)

To determine the important value of the factor *X* , one can employ a general formula,4

relating the relativistic mass *M* with the mass *M*0 at the origin,

2

0 _{2}

0

1 4 / (1 )

1 2 /

*M*

*r*

*r*
*dm*

*M* *X* *X* *M*

*Gm* *c r*

##

(12)5

The positive ratios *R r*/ * _{s}* from Eq. (11) and the real ratios

*M*0 /

*M*from Eq. (12) are plotted in Fig. 1. As is seen, there is no overlap between them for

*X*0. Moreover, the negative

*X*is always related to a negative mass defect, which indicates lack of bounded body for dark matter. Looking for a positive binding energy, the necessary inequality

*M*0

*M*imposes that

*X*0. If the mat-ter is super-relativistic with

*X*1/ 3, the corresponding mass defect is 24.4 %. If the particles are moving much slower than the speed of light, the doubled non-relativistic factor

*X*2 / 3 results in 28.6 % mass defect. The maximal mass defect about 29.3 % at

*X*1 should correspond to the lowest body temperature in order to prohibit any further energy loss by quantum reasons. It is less than 30 % and the matter particles are probably at rest. Logically,

*X*1 corresponds to the

minimal body radius 2

2* _{s}* 4 /

*R* *r* *GM c* from Eq. (11). According to Fig. (1), the radius of a
self-gravitating body is at least twice larger than the Schwarzschild radius.

**Fig. 2. The dimensionless potential ** 2

2 / *c* from Eq. (13) (solid) and Eq. (9) (dash) versus *r R*/

The finite value of the surface potential 2

ln 2 / 2

*R* *c*

follows from Eq. (9) at *R*2*rs*.
Hence, the overall potential (11) acquires the form, where *H* is the Heaviside step function,

2 2 2

6

Its plot in Fig. 2 shows lack of singularity in contrast to the Schwarzschild potential (9). The
po-tential (13) possesses a kink at *R*, which reflects the effect of a pressure jump at the body
sur-face. Hence, the body possesses a capillary pressure, 2 3

/ 4
*R*

*p* *Mc* *R* , which is related to the
surface tension via the Laplace law *p _{R}* 2 /

*R*

*. The latter can be integrated directly to*

_{R}obtain 2

/

*Mc* *A*

, where 2

4

*A* *R* is the area of the body surface. It is a particular example of
a size-dependent surface tension, described via the Tolman formula.5_{ It vanishes at a flat surface }

and the bending elasticity 4

2*R* *c* / 8*G* coincides with the universal Einstein expression.
Re-markably, the full energy 2

*Mc* *A* of the body is at the surface, which supports our expectation
that the body is energetically empty and that is why it cannot shrink anymore.

The problem now is what causes the pressure 2

*p* *c* if the particles are not moving. This
equation looks very similar to the ideal gas equation of state *p**Nk T V _{B}* / at constant
tempera-ture and we are going to explore such an identity. The characteristic potential of the body is the
Helmholtz free energy

*F T V N*( , , ) as a function of the natural parameters. Substituting the ideal gas pressure in the thermodynamic relation

*p* (

*VF*)

*T N*, allows direct integration to obtain

3

[ln(8 / ) 1 ( )]

*B* *P*

*F**Nk T* *l N V* *g T* (14)

where 3 1/2

( / )
*P*

*l* *G* *c* is the Planck length and *g* is an unspecified function. Using now the
equiv-alence of the pressure and energy density at *X* 1 determines the energy *E* *pV* *Nk TB* . In
the general case it reads *E**Nk T X _{B}* / . Substituting Eq. (14) into the thermodynamic relation

,
( * _{T}* )

_{V N}*E* *F T* *F* yields the unspecified function 2

ln( * _{P}* / 4

*)*

_{B}*g* *m c* *k T* , where 1/2

( / )
*P*

*m* *c G*

is the Planck mass. So, the free energy reads

2 2

[ln(2 / ) 1]

*B* *B*

*F**Nk T* *G* *N* *c k TV* (15)

Using this fundamental equation, one can calculate the chemical potential of the gas

2 ,

( * _{N}F*)

*2*

_{T V}*k T*ln( / 2

_{B}*k T*)

_{B}*k T*ln(8

_{B}*Gp c*/ )

7

gradients of the chemical and gravitational potentials cancel exactly in agreement with Eq. (10), 2

/ 2 / 0

*r* *k TB* *r* *c*

. Integrating the latter equation yields the gravito-chemical potential

2

2*k TB* /*c* 2*k TB* ln( *c*/ 4 *Rk TB* )

(17)

which is constant everywhere in the body. Since the body is energetically hollow, the
correspond-ing ( )*T* 0 determines the temperature of the body

/ 4 _{B}

*T* *c* *Rk* (18)

This is, in fact, the minimal energy-time Heisenberg relation. In black holes Eq. (18) reduces at
*s*

*R**r* to the Hawking temperature, *T _{H}*

*c*/ 4

*r k*

_{s}*.6*

_{B}_{ Therefore, due to }

_{2}

*s*

*R* *r* the temperature
of the body is half of the Hawking temperature. The relation 2

*Mc* *A* implies also that the body
entropy compensates exactly the negative pressure-volume term *S* *pV T*/ *NkB* to cancel the
internal energy completely. Substituting here the temperature (18) yields that the body entropy
coincides with the Bekenstein-Hawking one7

2 2

/ * _{B}* / 4

_{P}*S* *Mc* *T**k A* *l* (19)

Replacing this expression in the thermodynamic Maxwell equation *T* *AS* allows direct
in-tegration to obtain the temperature dependence of the surface tension 2

( ) / 4

*B* *c* *P*

*k T* *T* *l*

with

the critical temperature *Tc*. Substituting here

2

/

*Mc* *A*

and *T* *T _{H}* / 2 yields straightforward
that the critical temperature

*Tc*

*TH*is the Hawking one. Therefore, a black hole possesses zero surface tension 0, which means a zero mass

*M*.

The mass of a gas particle 2

/ * _{B}* /

*m**M N* *k T c* is extremely small at moderate
tempera-tures. Because any movement in the body is frozen, temperature causes solely some energy
fluc-tuation 2

*mc* with lifetime 2

/ 2*mc* 2 *R c*/

8

2 2

( / ) exp( / )

*m* *B* *B*

*f* *c* *k T* *mc* *k T* (20)

Perhaps, the neutrino is the only known particle possible to transmit the energy fluctuations in a cold body. Moreover, 2

/16

*P*

*m**m* *M* could be explained via the seesaw mechanism.8_{ From this }

perspective, it is interesting to consider a pure neutrino star. Measurements of the neutrino mass are very difficult, but important progress is achieved in the measurement of the difference in the squares of masses 2 2 2

12 1 2

*m* *m* *m*

via KamLAND.9_{ Assuming logically independent neutrinos }

in the ideal gas, it follows

1 2

2 2 2 2

12 1 2 1 2

0 0 *m* *m* 3

*m* *m* *m* *f f dm dm* *m*

##

(21)Using the experimental value 2
12 79
*m*

meV2_{/c}4_{, the corresponding temperature of the }

neu-trino star *T* 60 K is low enough to neglect any motion. One can estimate further from Eq. (18)
a very small radius *R*3 µm of the neutrino micro-star, while its mass 21

10

*M* kg is 6000 times
smaller than the mass of the Earth. The number of neutrinos in the star is 2 59

/ *B* 10
*N**Mc* *k T*

and from the volume per a particle 3 75

4 / 3 10

*v* *R* *N* m3_{ one can estimate the neutrino }

di-ameter as smaller than 25

10 m.

Obviously, the Fermi energy is huge due to the enormous density of the neutrino star and the tiny neutrino mass. This implies that the neutrinos form probably bosonic Cooper pairs (e.g.

majorons) to avoid the fermionic repulsion due to the Pauli principle. Perhaps, this is the reason
for the factor 2 in the gravito-chemical potential 2*m*. Thus, the neutrino star is a
Bose-Einstein condensate, since *T* is below the BEC critical temperature and 0. That is why the
body cannot emit energy anymore to collapse in a black hole. In a previous paper, we derived
that a self-gravitating quantum matter should always form a central hollow cavity.10_{ To examine }

this prediction, one can introduce now the exact relativistic density 2

/ 4

*M* *r R*

into the Bohm quantum potential to obtain

2 2 1/2 2 1/2 2

( / 2 ) *r*( *r* ) / ( ) / 16 ( ) /

*Q* *m* *r* *r* *r* *mr* *GM* *r* *cr* (22)

9 center.

The paper is dedicated to Stephen Hawking (1942-2018).

1. **R.C. Tolman, Phys. Rev. 55 (1939) 364 **

**J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55 (1939) 374 **

2. *K. Schwarzschild, Preuss. Akad. Wiss. (Berlin), Math.-Phys.-Tech. (1916) 189 *
3. **D. Finkelstein, Phys. Rev. 110 (1958) 965 **

4. *S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory *

*of Relativity, Wiley, New York (2013) *

5. * R.C. Tolman, J. Chem. Phys. 17 (1949) 333 *
6.

**S.W. Hawking, Nature 248 (1974) 30**7. **J.D. Bekenstein, Lett. Nuovo Cim. 4 (1972) 737 ****S.W. Hawking, Phys. Rev. Lett. 26 (1971) 1344 **

8. *M. Gell-Mann, P. Ramond, R. Slansky, in Supergravity, P. van Nieuwenhuizen, D.Z. Freedman *
(eds.), North Holland, Amsterdam (1979), p. 315